Research Interests 

My main research interests are in nonlinear elliptic and parabolic partial differential equations and applications in geometry and optimal transportation. In particular, the regularity theory of Monge-Ampère equations, Hessian equations, and other variational problems. 

I am also interested in related areas of geometry and physics, including geometry of convex bodies, minimal surfaces, surfaces of prescribed curvatures, and geometric flows. 

My research profile can be accessed via the following links:

Preprints

1. Optimal (partial) transport between non-convex polygonal domains, with S. Chen and Y. Li. Available at arXiv:2311.15655.

2. Global regularity in the Monge-Ampère obstacle problem, with S. Chen and X. Wang. Available at arXiv:2307.00262.

3. Regularity of singular set in optimal transportation, with S. Chen. Available at arXiv:2210.13841v2.

4. Regularity of  free boundaries in optimal transportation, with S. Chen. Available at arXiv:1911.10790.

5. Boundary regularity for the second boundary-value problem of Monge-Ampère equations in dimension two, with S. Chen and X.-J. Wang. Available at arXiv:1806.09482.

Publications (in mathematics)

1. $C^{2,\alpha}$ regularity of free boundaries in optimal transportation,  with S. Chen and X.-J. Wang. Communications on Pure and Applied Mathematics, Vol. 77, no. 1 (2024), 731--794. 

2. Regularity of optimal mapping between hypercubes, with S. Chen and X.-J. Wang. Advanced Nonlinear Studies, Vol. 23, no. 1, (2023) pp. 375--417. 

3. Preface to the special issue: nonlinear PDEs and geometric analysis - dedicated to Neil Trudinger on the occasion of his 80th birthday, with J. Clutterbuck. Mathematics in Engineering, Vol. 5, no. 6, (2023), Paper No. 095, 5pp. 

4. Non-uniqueness of solutions to the dual $L_p$-Minkowski problem, with Q.-R. Li and J. Lu. International Mathematics Research Notices, IMRN. (2022), no. 12, 9114--9150. 

5. Global regularity for the Monge-Ampère equation with natural boundary condition,  with S. Chen and X.-J. Wang. Annals of Mathematics, 194 (2021), 745--793.

6. A class of nonlinear optimisation and applications. Methods and Applications of Analysis, 28 (2021), 31--52.

7. Noncompact $L_p$-Minkowski problems, with Y. Huang. Indiana University Mathematics Journal, 70 (2021), 855--880.

8. Interior $C^2$ estimate for Monge-Ampère equation in dimension two. Proceedings of the American Mathematical Society, 149 (2021), 2479--2486.

9. Light refraction is nonlinear optimisation, with X.-J. Wang. Journal of Mathematical Study, 54 (2021), 142--163.

10. Optimal transport with discrete long range mean field interactions, with G. Loeper. Bulletin of Mathematical Sciences, 10 (2020), no. 2, 2050011, 26pp. 

11. The $L_p$-Brunn-Minkowski inequality for $p<1$, with S. Chen; Y. Huang and Q.-R. Li. Advances in Mathematics, 368 (2020), 107166, 21 pp. 

12. Stochastic continuity of random fields governed by a system of stochastic PDEs, with K. Du and F. Zhang. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 56 (2020), 1230--1250.

13. Global regularity of optimal mappings in non-convex domains, with S. Chen and X.-J. Wang. SCIENCE CHINA Mathematics, 62 (2019), 2057--2072. 

14. Bergman-Toeplitz operators on fat Hartogs triangles, with T.-V. Khanh and P.-T. Thuc. Proceedings of the American Mathematical Society, 147 (2019), 327--338. 

15. Bergman-Toeplitz operators on weakly pseudoconvex domains, with T.-V. Khanh and P.-T. Thuc. Mathematische Zeitschrift, 291 (2019), 591--607. 

16. On the Cauchy problem for stochastic parabolic equations in Hölder spaces, with K. Du. Transaction of the American Mathematical Society, 371 (2019), 2643--2664. 

17. Optimal transport with discrete mean field interaction, with G. Loeper. 2017 MATRIX Annals, Editors: David R. Wood, Jan de Gier, Cheryl E. Praeger, Terence Tao. MATRIX Book Series, Volume 2, Springer, pp.207--212, 2018.

18. Schauder estimates and application on stochastic partial differential equations. Surveys in Geometric Analysis 2017, Editors: Gang Tian, Qing Han, Zhenlei Zhang, Science Press, pp.94--115, 2018.

19. A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions, with Y.Y. Li and L. Nguyen. Journal of Fixed Point Theory and Application, 19 (2017), 853--876.

20. A Schauder estimate for stochastic PDEs, with K. Du. C. R. Math. Acd. Sci. Paris, Ser. I, 354 (2016), 371--375.

21. On the classical solvability of near field reflector problems, with N.S. Trudinger.  Discrete Contin. Dyn. Syst., 36 (2016), 895--916.

22. On the uniqueness of $L_p$-Minkowski problems: The constant $p$-curvature case in $\mathbb{R}^3$, with Y. Huang and L. Xu. Advances in Mathematics, 281 (2015), 906--927. Available at arXiv:1503.02358.

23. Boundary $C^{2,\alpha}$ estimates for Monge-Ampère type equations, with Y. Huang and F. Jiang. Advances in Mathematics, 281 (2015), 706--733.

24. On asymptotic behaviour and $W^{2,p}$ regularity of potentials in optimal transportation, with N.S. Trudinger and X.-J. Wang. Arch. Rat. Mech. Anal., 215 (2015), 867--905.

25. Interior a priori estimates for the Monge-Ampère equation, with X.-J. Wang. Surveys in Differential Geometry Volume 19, International Press (2014), 151--177.

26. Optimal transportation on the hemisphere, with S.-Y. A. Chang and P. Yang. Bulletin of the Institute of Mathematics, Academia Sinica 9 (2014), 25--44.

27. Degree theory for oblique boundary problems, (with Y. Y. Li and L. Nguyen). Oberwolfach Report 40/2013:26--29.

28. An obstacle problem for a class of fourth order equations, with B. Zhou. Journal of Differential Equations 254 (2013), 1306--1325.

29. Light reflection is nonlinear optimization. Calc. Var. Partial Differential Equations 46 (2013), 861--878.

30. Regularity of Monge-Ampère equations in optimal transportation. Bull. Austral. Math. Soc. 83 (2011), 173--176.

31. On Pogorelov estimates for Monge-Ampère type equations, with N.S. Trudinger. Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 1121--1135.

32. Interior $C^{2,\alpha}$ regularity for potential functions in optimal transportation, with N.S. Trudinger and X.-J. Wang. Comm. in Partial Differential Equations 35 (2010), 165--184.

33. Regularity in optimal transportation, (with N. S. Trudinger and X.-J. Wang). Oberwolfach Report 36/2009:17--20.

34. Hölder regularity of optimal mappings in optimal transportation. Calc. Var. Partial Differential Equations 34 (2009), no. 4, 435--451.

Publications (in computer science)

1. A multiagent-based domain transportation approach for optimal resource allocation in emergency management, with J. Zhang; M. Zhang and F. Ren. Multi-agent and Complex Systems, pages 19--32, Springer, Singapore, 2017.

2. A decentralised multi-agent system for emergency resource allocation in metropolitan regions, with J. Zhang; M. Zhang and F. Ren. Proceedings of the 2016 International Conference on Autonomous Agents & Multiagent Systems, pages 1482--1484, 2016. 

3. Enable efficient resource deployment in multiple concurrent emergency events through a decentralised MAS, with J. Zhang; M. Zhang and F. Ren.  Australasian Joint Conference on Artificial Intelligence, pages 56--68, 2016.  

4. The singularity set of optimal transportation maps, with Z. Luo; W. Chen; N. Lei; Y. Guo; T. Zhao and X. Gu. Computational Mathematics and Mathematical Physics 62, 1313--1330, 2022.